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In a triangle, two sides have a combined length of x and their product is y. If the equation x² - c² = y holds true, where c represents the third side, what is the ratio of the triangle's in-radius to its circum-radius?

1. 3y / 2x(x + c)
2. 3y / 2c(x + c)
3. 3y / 4x(x + c)
4. 3y / 4c(x + c)

Correct answer: 3y / 2c(x + c)

Solution

Using the given equation x² - c² = y and properties of triangles, the ratio of the in-radius to the circum-radius simplifies to 3y / 2c(x + c). This result comes from the relationship between the sides, area, and radii of the triangle.

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